Emergence of digit-like structures through
mechanical instabilities:
A hybrid computational and experimental study
Steffen Plunder
Seirin-G
ASHBi Fusion Grant with
Rio Tsutsumi & Antoine Diez
ASHBi Retreat 2025
Big Picture
Regenerative medicine:
"replacing, engineering or
regenerating cells, tissues or organs
to restore or establish normal
function"
This project:
study of growth of limb bud and digit formation → tissue with a shape
Sullivan, A. E. & Santos, S. D.
The ever-growing world of gastruloids: autogenous models of
mammalian embryogenesis.
Current Opinion in Genetics & Development (2023).
Morphology:
How Does Nature Create Shapes?
How do cells organize themselves into the requires shapes?
Morishita, Y. et al. An archetype and scaling of developmental tissue dynamics across species.
Nat Commun (2023).
Morphology:
How Does Nature Create Shapes?
Is everything determined by morphogens?
Gilbert, S. F. & Barresi, M. J. F.
Developmental Biology.
(2018).
AER
(apical
ectodermal ridge)
How are
complex
shapes
like digits established?
Complex Shapes & Instabilities
Often symmetric states are stable...
However, if they are
instable
, this can break
symmetry and create complex shapes.
There are many types of instabilities in nature:
Resonance
Fluid interface
(Saffman–Taylor instability)
Ma, X., Zhong, M., He, Y., Liu, Z. & Li, Z.
Fingering instability in Marangoni
spreading on a deep layer of polymer solution.
Physics of Fluids (2020).
Turing instability
(diffusion based)
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Walker, B. J., Townsend, A. K., Chudasama, A. K. & Krause, A. L.
VisualPDE:
Rapid Interactive Simulations of Partial Differential Equations.
Bull Math Biol (2023).
Current state:
We know many details about the signals around digit formation!
Raspopovic, J., Marcon, L., Russo, L. & Sharpe, J.
Digit patterning is controlled by a
Bmp-Sox9-Wnt Turing network modulated by morphogen gradients.
Science (2014).
Required model
inputs:
Fgf signal
from the AER
Turing model (with pattern via diffusion based instability)
Current gap in existing models:
How do signals and tissue growth interact?
Our Approach
Limb Bud
in vivo
Turing Model
input: shape growth + signals
output: signal pattern
Agent-Based Models
input: signals + pattern
output: shape growth
complexity
math → biology
Digit Organoid
in vitro
digit formation
via
in vitro
model
gives access to more data!
reduced number of
mathematical
assumptions
Our Hybrid Model
input: experimental condition
output: shape growth + pattern + signals
Data Analysis
3D tracking data
Propose Mechanism
Key finding from modelling:
Formation of digit-like structures in organoid
could be explained by an interplay of
a
new mechanical instability
(not Turing) and
self-organized Wnt gradient.
Construction of a
Mathematical Model for
Digit Organoids
Digit Organoid
Rio Tsutsumi
Distal Cells
Proximal Cells
mix
Serum
Fgf8+Wnt5a/Serum
9000 cells
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Fgf8+Wnt5a/Serum
27000 cells
full in vivo limb bud has ~200.000 cells
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Note: No signaling center
& no AER!
Question: How can
proximal
&
distal
cells self-organize
i)
sorting
ii)
symmetry breaking &
iii)
elongation/digit-formation?
Fgf+Wnt/Serum
We use mathematical modelling to test how this could be explained.
How do cells move?
Analysis
of cell tracking data
Core modelling question:
How can an organoid elongate?
Micromass experiments:
Evidence for chemotaxis
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distance
of attraction
~ 300µm
Indicates potential role of
morphogens.
Candidate elongation mechanism:
Planar Cell Polarity (PCP)
Vangl2 polarization shows potential PCP
Direction of extension
PCP leads to
convergent-extension
Destabilisation of adhesive
interaction at
Vangl/Fzd interfaces
Wnt can induce PCP:
Well-known role of Fgf/Wnt
gradient in limb bud:
In vivo evidence of Vangl2 in limb bud:
Proxiaml - Distal axis
todo: add references
The challenges of measuring converent-extension in vivo
Accumulation effect:
time
velocity
large velocity at the tip
is misleading!
Related data
Morishita, Y. et al. An archetype and scaling of developmental tissue
dynamics across species.
Nat Commun (2023).
(ex-plants)
PIV
Parada, C. et al.
Mechanical feedback defines organizing centers
to drive digit emergence.
Developmental Cell (2022).
The Green strain tensor:
A way to measure deformation without bias & accumulation effects!
Pipeline:
0. raw
3D tracks
2. interpolate velocity field
(Gaussian kernel smoothening)
1. cleanup
3. compute deformation map
following interpolated velocities
4. compute at each point the
Green strain tensor
frame 1
frame 30
deformation
map
Main idea:
Track how local sphere would deform within the tissue
(+ tiny amount of math to make it less sensitive to noise/rotations)
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Convergent-Extension Index
Elongation is drive by convergent-extension
Green strain tensor
Localization of
convergent extension
at distal-proximal
interface
Construction of a mathematical
model
Related works: Mathematical models for elongating organoids
Okuda, S. & Sato, K.
Polarized interfacial tension induces collective
migration of cells, as a cluster, in a 3D tissue. Biophysical Journal
(2022)
.
Vertex-based
Sphere-based
Okuda, S., Miura, T., Inoue, Y., Adachi, T. & Eiraku, M.
Combining
Turing and 3D vertex models reproduces autonomous multicellular
morphogenesis with undulation, tubulation, and branching.
Sci Rep
(2018)
.
Germann, P., Marin-Riera, M. & Sharpe, J.
ya||a: GPU-Powered
Spheroid Models for Mesenchyme and Epithelium.
cels (2019).
Up to our knowledge:
No hybrid (signal/mechanical) model
for elongating organoid
of mesenchymal cells
or limb buds.
Initial agent-based
organoid model
We describe each cell by a
- position
(in 3D space)
-
radius
(spherical cell model)
-
celltype
(proximal or distal)
N = 10.000 ~ 30.000
Initial position & cell type are random.
Rules
Repulsion
Differential
Adhesion
Experimental
data
angle
cortical
tension
adhesive
force
stronger
adhesion force →
Newton's 3rd Law
With structurally stable ECM
Force balance
cell forward
pulling on ECM
Basic Physics: How can cells move forward?
Organoid setting
(no structurally stable ECM
in matrigel)
#1
#2
#3
pulled back by #2
pulling on #3
pulling on #2
pulled back by #1
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proximal
/
distal
✔ sorting
✘ sym break
✘ elongation
Optional math:
All three cases are included in a simple
(symmetric) force function:
(direction of adhesion)
(signal direction)
(aligment of signal & direction of force)
Planar Cell Polarity (PCP)
between distal cells
Wnt signal
distal
-
distal
primary lateral adhesion
Convergent Extension
Wnt
Chemotaxis of
distal
cells
through
proximal
cells
distal
-
proximal
Wnt signal
chemotaxis via biased adhesion
Chemotaxis
Wnt
Proximal
cells ignore
morphogen
front
lateral
rear
cell-cell adhesion
forces
D-D
P-P
D-P
A minimal mathematical model for PCP and more...
GPU-accelerated simulations
for realistic cell counts (>10.000 particles)
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→ github.com/antoinediez/Sisyphe
Wnt
(more complex examples later...)
Convergent-Extension
Unexpected
consequence of PCP:
a new mechanical instability...
Recall: without PCP/Wnt gradient
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Assume there would be an radial Wnt gradient field...
Wnt
Illustrative 2D example:
Key modeling insight:
PCP leads to an mechanical instability which leads to digit-formation.
Establishing a Wnt gradient
via a simple pathway
How could a Wnt gradient be established?
Digit Organoid
Fgf+Wnt/Serum
Known:
1. Fgf
(in cell junctions):
inside:
short range diffusion + fast decay
outside:
available en mass
2. Wnt
(membrane bound):
inside:
medium range diffusion + slow decay
outside:
[irrelevant for math model]
Fgf8 in
aggregate
dpERK FGF pathway
activity
Ten Berge, D., Brugmann, S. A., Helms, J. A. & Nusse, R.
Wnt and FGF signals interact to coordinate growth with
cell fate specification during limb development.
Development (2008).
3. Fgf signal triggers
distal
cells to emit Wnt
Hybrid Agent-based + Reaction Diffusion Model
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Hybrid model explains possible mechanism for
i)
sorting
ii)
symmetry breaking
& iii)
elongation/digit-formation!
Symmetry breaking via a feedback between cell dynamics and morphogenes
Symmetric start:
- spherical organoid
- radial symmetric signal
distal
/
proximal
Fgf
Wnt
Symmetry breaking:
-
Mechanical instability:
leads to asymmetric distal cell clusters
- feedback with morphogens
via
Fgf+distal-Wnt pathway
Fgf
Wnt
Established Wnt gradient:
- Distal cells keep emitting Wnt,
maintaining distal-proximal
Wnt gradient.
- Proximal cells do nothing, but
are essential to create Wnt gradient!
Fgf
Wnt
Comparison with experimental data
Simulations recapture behaviour at different ratios of distal cells:
Digit Organoid
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Simulations
Ratio of distal cells within organoid
(higher total cell count)
Green strain tensor
Convergent-Extension Index
3D tracking data: Digit Organoid
Our hybrid model (in silico)
Localization of
convergent extension
at distal-proximal
interface
Conclusion
Mathematical model suggests:
Digit-formation might
not reply only
on
Turin
instability (diffusion based)
but also
use a
mechanical instability
!
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Comparing simulations and digit organoid data
suggest that
PCP + Fgf-Wnt crosstalk
creates
digit-like elongations.
Supporting the idea
digit organoid
elongations
recaptures in vivo mechanism.
Closing remarks from a mathematical point of view
Using statistical physics,
Antoine could derive the underlying
equation for densities of distal cells:
(much easier to analyze,
finding concepts)
(good for computers,
bad for theorety/insights.)
Cahn-Hilliard PDE
Open mathematical
questions to characterize instability
Work in progress:
Antoine Diez & Hyunjoon Park
Equation for fluid interfaces:
gas
oil
Ye, Y. & Lin, J.
Fingering Instability Accelerates
Population Growth of a Proliferating Cell Collective.
Phys. Rev. Lett. (2024).
Recently also studied
for tumor growth:
Physicists called this type
"fingering" instability.
(Up to our knowledge, this type of instability was never been linked to digit/finger growth so far.)
BiMed-Math Group
Sungrim Seirin-Lee
Antoine Diez
Tsubasa Sukekawa
Ying Xie
Jinghao Chen
Tamaki Wakamoto
Michito Ujino
Takahiro Hiraga
Chie Shirai
ASHBi
Support and
nuturing scienfic
exchange:
Makoto Shida
Fumi Komori
Spyros Goulas
All members of Alev-G
Takafumi Ichikawa
Rio Tsusumi
Antoine Diez
Thank You
ASHBi Fusion Grant
EdU test:
Proximal
and
distal
cells
proliferate at a similar rate.
Fgf/Wnt
We assume proliferation has no
major effect within digit organoid.